I don't like the idea of moving the swipe area a little higher because moving it a little doesn't really solve the problem. Currently the swipe area covers the whole toolbar. So it should be moved all the way above the toolbar buttons which makes toolbar dismissal very awkward. You would need in that case be very precise where to grab the toolbar and in addition there is no visual cue where the grab area is. Of course one option is to increase toolbar size and add the grab area. But visually it's not very appealing.

The current solution is quite nice but obviously has its own issues now when it misinterprets some of the user's gestures. But there is room for improvement I think. A threshold value. Toolbar shouldn't start dragging until certain threshold value is passed. Clearly simple click of a button doesn't pass the value and the buttons would get their click event properly.

Either way this should probably have some consistency with the OSK. So, the toolbar could only accept input just above itself like the keyboard, or the keyboard can go back to dragging anywhere on the keyboard.

The keyboard changed behaviour to solve the exact same problem, so it makes sense that both components should use the same solution (whichever solution that is), which also makes it easier for the user to know what to expect if all components behave similarly.